You can figure this out by looking at anyone else's profile. The fields which are displayed publicly (and thus accessible to search engines), if you have filled out the corresponding parts of your profile, are:
- website
- location
- age (which is calculated from your birthday)
- bio information (the "About Me" box)
Naturally, you can decline to enter anything for the website, location, and birthday fields, or in the "About Me" box, in which case the corresponding parts of your profile will be left blank.
Other information displayed on your profile includes:
- member for (how long you've been a member)
- seen (when you last visited the site under that account)
- profile views
- links to all non-deleted questions and answers you've posted
- recent activity, including questions, answers, and comments posted, edits submitted or reviewed, answers accepted, badges earned (but not votes cast)
- reputation earned (but not from downvotes you cast)
- tags on questions you've asked or answered
- badges you've earned
- favorites you've marked
- bounties you've offered or been awarded
- accounts on other SE sites that you've associated with the current one
- total numbers of upvotes and downvotes cast, not associated with individual questions or answers
The remaining fields of your profile are not visible to anyone except moderators, who are forbidden from revealing them publicly under the terms of the moderator agreement. In particular:
- real name
- email
- birthday (the year can be calculated because your age is displayed, but the actual date is hidden)
Also, your exact voting record (which posts you've voted for and what vote you cast for them) is not visible to anyone, not even moderators.
As far as privacy is concerned: nothing on your profile allows your Stack Exchange account to be linked with your real identity, unless you explicitly put personally identifying information in the public fields or in your posts. (I'm not sure what the point of changing your username each month is, but it's fine if you want to do it...)